## FANDOM

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The q-factorial is the q-analog of the factorial function.[1] It is written $$[n]_q!$$ or $$\mathrm{faq}(n,q)$$ and is defined as

$[n]_q! = \prod^{n - 1}_{i = 0} \left(\textstyle\sum^{i}_{j = 0} q^j\right) = q^0 \cdot \left(q^0 + q^1\right) \cdot \left(q^0 + q^1 + q^2\right) \cdot \ldots \cdot \left(q^0 + q^1 + \ldots + q^{n - 1}\right)$

As with all q-analogs, letting $$q = 1$$ produces the ordinary factorial.

Based on the q-factorial, we can define the q-exponential function:

$e^x_q = \sum_{i = 0}^{\infty} \frac{x^i}{[i]_q!} = \frac{1}{[0]_q!} + \frac{x}{[1]_q!} + \frac{x^2}{[2]_q!} + \frac{x^3}{[3]_q!} + \cdots$

as well as q-trigonometric functions $$\sin_q x = \frac{e^{ix}_q - e^{-ix}_q}{2i}$$, $$\cos_q x = \frac{e^x_q + e^{-x}_q}{2}$$, etc.

### Values

1 2 3 4
1 1 1 1 1
2 2 3 4 5
3 6 21 52 105
4 24 315 2,080 8,925
5 120 9,765 251,680 3,043,425

1. [1]